Anti-Bernouilli Random Variable, Time Reversal Invariance and the Quantum exp(-Et)

Ant-Bernouilli Random Variable, Time Reversal Invariance and the Quantum exp(-Et) Francesco R. Ruggeri Hanwell, N.B. Nov. 6, 2025 In (1), a type of probability conservation of an anti-Bernouilli variable linked to diagonal motion along box diagonals in an x-t space with 1-mdt being the probability that x direction motion changes leads to a probability of exp(imt). This approach seems to involve a kind of time reversal symmetry as (1) proposes a t-x space consisting of grids of spacing e and possible motion along diagonals with both +/- e motion in t and x. In particular, (1) considers motion in the positive t direction from the origin to a some T (with +/-e motion possible) and then retraces the direction to return to the origin. In other words, (1)’s m is not an arbitrary number, but is linked to a conserved quantity, energy, i.e. probability steps are in terms of energy if one compares with the quantum exp(-Et), which (1) does. (1) obtains a kind of Dirac equation, but although we provide analysis of this, we only consider the form exp(imt) as relevant. Furthermore, exp(iEt) is consistent with conservation of energy and conservation of momentum follows from the x form of the solution exp(ipx) and so the probability approach of (1) is somehow linked to a change in direction, i.e. collision, which ultimately is associated with a conserved m, i.e. E (energy). In previous notes, we suggested that one may obtain the free particle quantum wavefunction exp(-iEt+ipx) by suggesting that a probability exists in Newtonian two-body elastic scattering. We argued that given initial (e1,e2) and (p1,p2) (energy, momentum) particles, any (ei,ej) (pi,pj) vectors has the same product probability (equal to the product probability of the (e1,e2) (p1,p2) case) has the same value. In (2), we showed that one cannot use exp(ip) for the momentum probability because it is not unique and violates time reversal symmetry. We suggested using a Lorentz invariant result exp(-iEt+ipx). The argument we make in this note is that the approach of (1) and (2) seem to be the same. At first this seems to be surprising. (1) obtains exp(-iEt) strictly through a “special” anti-Bernouilli somewhat-random walk approach. There is no concept of energy or momentum, hence no concept of their conservation and no concept of special relativity. (2) on the other hand is based on all of these ideas. We suggest that given that probability is introduced in (2), the notion of physical paths and changes is common to both approaches. Furthermore, we suggest that both are based on time reversal symmetry. We have already noted that the number m in (1) is equivalent to energy. Thus, energy is linked with the anti-Bernouilli probability motion in time (and p with motion in x). This, in a sense, is the same as a conservation of energy and momentum approach, we argue. If one considers that energy and momentum follow from special relativity by Lorentz boosting a rest mass, then one really has the concept of conservation of the rest mass which is the same as conservation of a certain kind of probability. Thus, we argue that (1) is an interesting approach to flushing out the probabilistic nature of the Dirac equation which we argue is also directly linked to introducing a dynamic directional probability which describes energy and momentum conserving interactions. Anti-Bernouilli Quasi-Random Walk of (1) In (1) a special probabilistic quasi-random walk which leads to the derivation of an exp(imt) probability is introduced. The key point we make is that this is a “special” type of probability. In (1), a standard Bernoulli approach is used and it does not lead to exp(imt), but to (cosh(mt), sinh(mt)). As a result, we suggest one must analyze the assumptions which are used in (1)’s probabilistic model carefully, we suggest. A t-x grid is considered with spacing of e. A particle may move to the right or left, but only along a diagonal. If a particle moves to the right in x, this is associated with a function phi+(x,y), otherwise with phi-(x,y). A particle which has just moved to the right has Probability 1-em to move to the right on the next move ((1)) Probability em to move to the left ((1b)) and vice versa Thus, a key idea is introduced. The number m governs the probability of motion in x. An anti-Bernouilli variable is introduced as: Value = 1 if the particle moves from t to t+dt ((2)) 0 if it doesn’t move -1 if it moves backwards in time from t to t-dt In (1) it is stated that a particle begins at the origin and moves only in the positive time (t) direction until it reaches a time T. At each point, it may move in the positive or negative x direction according to ((1)). At every other point where em applies, i.e. it should change direction, it doesn’t, but places a marker which may be used on the return trip. As a result, (1) provides zigzag diagrams of motion in the positive time direction and the in the negative time return trip. These two zigzags create rectangles. If one chooses a path along the outside walls of the rectangles, one obtains a single path which contains both forward and negative times. It is these paths which (1) considers and he weights negative time paths with a -1. This is his anti-Bernouilli variable. In (1) it is stated that one may define two functions, phi+(x,t) and phi-(x,t) to account for the average of the anti-Bernouilli variable ((2)) in either the positive or negative x direction. Then: Phi+(t) = Sum over x phi+(x,t) and Phi-(t) = Sum over x phi-(x,t) ((3)) In (1) it is stated that to conserve probability one has: Phi+(t+e) = (1-em) Phi+(t) - em Phi-(t) ((4a)) Phi-(t+e) = (1-em) Phi-(t) + em Phi+(t) ((4b)) We wish to examine ((4)) in more detail. First, one notices that ((4a)) contains a minus sign and ((4b)) does not. Thus, there is symmetry breaking. To try to see this result, we follow diagrams provided by (1). We first consider zigzag motion in x, but only an increase in t to a certain T. This zigzag is coloured blue and represents forward motion in time. Then, one creates a red zigzag making rectangles with the blue zigzag. This represents return motion which is negative in time (i.e. -1 Bernouilli value). If one considers two blue outer rectangle lines, then an outer red line emanates perpendicular to the second. In other words one has a zigzag of blue (positive x direction)-blue (negative x direction)-red (positive x direction). We stress that the zigzag is a combination of forward and negative time motion, i.e. it is a mathematical path which consists of pieces of a forward in time random type walk and a negative. The second junction shows blue x-neg and red-x positive or Ph-(t+e) = (1-em) Phi-(t) -1 Phi+(t) The first two junctions of the zigzag (blue positive x) - (blue negative x) yield: Phi+(t+e) = Phi+(t) (1-em) + em Phi-(t) This is the opposite of ((4)), but that is not a problem. The point is that one equation should have a positive sign and the other, not. This leads to: Phi= (Phi+, Phi-) and d/dt Phi = -m Phi -m i Pauli sigma(y) ((5)) With Phi+(0)= Phi-(0)=0 one finds that Phi+(t)= exp(-mt) (cos(mt), sin(mt)) ((6)) The key point of ((6)) is that not one has the form exp(imt) which is a quantum free particle result linked with conservation of m. Thus, conservation of m naturally appears. In fact, this mimics the quantum exp(-iEt) and for t=dt, exp(-iEdt) = 1-Edt which is the probability (1) uses for a direction change in x not to occur. In other words exp(-iEdt) means that the particle continues in its same general x direction for a free non-interacting particle. One must, however, account for x as well t. In (1), this is done through putting in the x detail in the two zigzags described above. (Note: The zigzags describe interchange Phi+ and Phi-, but that is irrelevant.) Given that x is present, we now use the notation phi+(x,t), phi-(x,t) as Phi+ means one has summed over x. phi+(x,t+e) = (1-em) phi+(x-e,t) - em phi-(x,t) ((7a)) We note that the second term on the RHS is phiwhich changes its direction. phi-(x-e, t+e) = (1-em) phi-(x,t) + em phi+(x-e,t) ((7b)) ((7a)) and ((7b)) are expanded in a Taylor series. We obtain a different sign in one term than (1) and so write ((8)) (see below) as -sigma Pauli z not Sigma-z Pauli. ((8)) is written in the form of ((1)). Then (1) calls phi+/- = exp(-mt) W+/- and obtains the key equation: d/dt W = sigma Pauli z d/dx W -im sigma Pauli y W ((8)) In (1), ((8)) is called a Dirac equation although formally the Dirac equation involves 4x4 matrices and 4-vectors. If one wishes to write it in terms of a 2-vector, then one has a mixing of the two 2-vectors comprising the 4-vector, so we are uncertain as to how ((8)) constitutes a Dirac equation. A second problem is that sigma-z and sigma-y arise naturally, but it is not clear what to call the variable x. It could be x,y,z. Finally, we suggested that m acts as energy in exp(i m t), but in ((8)) it has the appearance of a kind of rest mass which is confusing. We thus limit ourselves to the form exp(imt) =exp(iEt) from the analysis of (1). This also has an issue because there is a damping exp(-mt) term as well as the two-vector cos(mt)+i sin(mt), but we focus on the latter. Comparison with Newtonian Scattering Probability In previous notes (2), we argued that one may introduce a free particle probability into Newtonian 2-body scattering. In such a case, an initial (e1,e2) energy and (p1,p2) momentum vector sets should give rise to possible outcomes (ei,ej) (pi,pj) each of which have the same probability if momentum and energy are conserved. Given time reversal invariance, we argued that one cannot use exp(ip), but should use exp(ipx). This is also necessary because exp(ip) is not unique because p1=p+ 2n*3.14 has the same probability. Thus, as noted in (1), it is a wavelength (i.e. a spatial frequency) which gives uniqueness). We finally settled for: exp(-iEt+ipx) ((9)) which is Lorentz invariant. One may argue that the above approach requires knowing about momentum and energy as well as time-reversal and Lorentz invariances and conservation of energy and momentum. We suggest that time-reversal invariance for exp(ipx) i..e p→-p x→-x is contained in Lorentz invariance and Lorentz invariance also respects conservation of momentum in the sense. First, given a rest mass mo at x, the notion of momentum and energy follow from observing this object from a frame moving with constant -v. This is an alternative approach to Newton’s consideration of a force acting on mo. In a center-of-mass frame, all momenta add to zero and so one in a sense has an overall mo with ptotal=0. If the center-of-mass frame is not accelerated, i.e. all forces are action-reaction internal ones, then one is guaranteed momentum conservation. Thus, we argue that it is no accident that ((9)) which is Lorentz invariant has built-in momentum and energy conservation. We obtained it here by using conservation a priori, but in other notes we derived it without any consideration of conservation, but only by examining the Lorentz invariant: -Et+px = constant ((10)) The point we make is that one uses an invariance idea (Lorentz invariance) and finds that it gives rise to conservation of energy and momentum. We thus suggest that assumptions in the probability model used by (1) may also lead to conservation of energy, which (1) finds. In particular, the approach of (1) seems to be linked to a kind of time reversal symmetry as (1) allows for a zigzag path in x in positive time and then reverses the path to return to the origin. He then analyzes the two paths in terms of rectangles, creating a new path based on the outside rectangular pieces of each and applies a conservation of probability to this new “created” path. He gives a weight of -1 to pieces which represent travel backwards in time (i.e. the return trip). Furthermore, he gives a weight for a change in direction in x as: m dt ((11)) Here m is some number. The probability to continue in a certain direction is then: 1mdt ((12)) This is identical to exp(-Et) for t=dt. In other words, conservation of probability is linked to a variable m which acts like E (energy). The approach of (1) does not only conserve probability, but finds a probability expression which conserves m, the variable linked to changes in direction of x, i.e. part of the probability conservation equation. Thus, we think that the ideas of (1)’s probability approach are very similar to those used in creating an exp(-iEt+ipx) probability based on time reversal invariance for exp(ipx) and overall Lorentz invariance and we suggest that the approach of (1) may be used as an example of probability used to obtain exp(iEt). We note that changing the direction of motion in x is physically equivalent to scattering which is what the exp(-iEt+ipx) approach is all about-scattering with conservation of E and p and a real weight of 1 for any free particle. Conclusion In conclusion, we suggested in previous notes (2) that one may obtain the quantum free particle probability exp(-iEt+ipx) by considering that a given set (e1,e2) energy (p1,p2) momentum vectors in Newtonian mechanics has equal weight for any outcome set (ei,ej) (pi,pj) if energy and momentum are conserved. At first, one might think this leads to exp(-E C1) exp(p C2) (constants for units), but neither of these are unique as p1=p+2*3.14*n (and a similar expression) are the same. One wishes to have a unique probability. Furthermore, exp(pC2) is not time reversal invariant. To make it so, one might introduce exp(ipx) and finally exp(-iEt+ipx) which is Lorentz invariant. This gives a probability linked to change in time as exp(-iEt) which becomes 1-Edt for t=dt. Thus, there is a free particle energy conserving probability for tiny dt steps linked to 1-Edt. In (1), an anti-Bernouilli variable sort-of random walk is introduced which also takes into account time reversal. A particle starts at the origin and moves along the diagonal of squares in and x-t space. It may move to the right or left in x, but only positive in dt until T is reached and then it reverses its motion (based on markers put down). (1) develops a conservation of probability scheme for paths which include both motion positive in t and negative and bases a change in x direction to a probability mdt with 1-mdt being the probability that there is no change. Such a change in x is linked with a collision and we note the similarity between 1-mdt and exp(iEdt). In fact, (1) finds that his anti-Bernouilli probability is proportional to exp(imt) and we suggest that this may be linked to our approach of introducing exp(-iEt+ipx) to a free particle probability which conserves energy and momentum. We note that a priori one does not need to know about conservation of energy and momentum because these follow from Lorentz invariance. In other words, one might find exp(-iEt+ipx) from the Lorentz invariant -Et+px = constant. A free particle can only have a unit real weight and so its probability should be complex, but exp(iEC1), exp(ipC2) are not unique (i.e. p1=p+2*3.14*n gives the same value). Thus, one requires two variables to have uniqueness and time reversal invariance. As a result, we suggest that some symmetry properties (time reversal or Lorentz invariance) may “bring in” physical conservation of energy and momentum. We suggest that the probability model of (1) does something very similar. References 1. Ord, G. Quantum Mechanics in a2-Dimensional SpaceTime-What is a wavefunction? (2009 or later) https://math.torontomu.ca/~gord/abstracts/QWave/QWave4ElsartR1.pdf 2. Ruggeri, Francesco R. Classical Elastic Scattering Probabilities and the Necessity of a Wavelength (preprint, zenodo, 2025) 3.https://en.wikipedia.org/wiki/Dirac_equation https://en.wikipedia.org/wiki/Pauli_matrices